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Burnley manager Sean Dyche fielded a largely second-string lineup, making eight changes from the team which drew with Brighton & Hove Albion in the Premier League on Saturday.
Bournemouth looked hungrier from the outset and caused Burnley plenty of problems before Surridge put them in front, slotting home from close range after a low ball across from Jack Stacey.
The Clarets created little but Jay Rodriguez should have brought them level in the 56th minute when he missed a glorious chance, scooping the ball over the bar from close range.
Former Burnley winger Stanislas made sure of the win, two minutes from the end, with a well-taken penalty after Kevin Long brought down Surridge in the box.
Bournemouth are into the FA Cup sixth round for the first time since 1956-57 and the win was a boost for interim manager Jonathan Woodgate who took over last week following the sacking of Jason Tindall.
“We wanted it but you need the players to execute it and they did that. I thought every single one of them was very good. I think that is the second time in our history we have reached the quarter-finals so fair play to the players,” said Woodgate, who declined to discuss his future.
“It is one game at a time for me. If I am in charge for the Nottingham Forest game [on Saturday] then I will do it to the best of my ability,” he said.